In the paper we study defined on an interval continuous functions where the argument and the values are represented ($G_2$-representation) in a coding system with two oppositely signed bases $g_0 \in [0,5;1)$ and $g_1=g_0-1$ and the two-symbol alphabett $A=\{0;1\}$:

$x = α_1g_{1-α_1} + \sum_{k=2}^∞ (α_kg_{1-α_k} \prod_{j =1}^{k-1} g_{α_j}) \equiv Δ^{G_2}_{α_1α_2...α_n...}.$

These functions are divided into three distinct classes. The first class includes functions defined by an equation:

$φ (x = Δ^{G_2}_{α_1α_2...α_n...}) = Δ^{G_2}_{r_1(α_1)r_2(α_2)...r_n(α_n)...},$

where $(r_n)$ is a given sequence of functions $r_n: A\to A$.We prove that in this class there exist no any continuous functions except constants, the identity transformation of the interval, and the function

$f (x = Δ^{G_2}_{α_1α_2...α_n...}) = Δ^{G_2}_{[1-α_1]α_2...α_n...}$

The second class is represented by the following functions:

$g (x = Δ^{G_2}_{α_1α_2...α_n...}) = Δ^{G_2}_{d(α_1,α_2) d(α_2,α_3)...d(α_n,α_{n+1})d(α_{n+1},α_{n+2})...},$ where $d: A × A → A.$

We prove that this class contains only four continuous functions: two constant functions, the identity transformation of the interval, and the left-shift operator for the digits of the $G_2$-representation of numbers. The third class consists of continuous strictly increasing singular functions (whose derivative is zero almost everywhere in the sense of the Lebesgue measure), defined by a system of functional equations:

\begin{cases} f (g_0x) = q_0f(x),  \\ f (g_0 + (g_0-1) x) = q_0 + (q_0-1)f(x), & {       }  q_0 ∈ [0,5;1) q_1 = q_0 -1 .\end{cases}The graphs of functions in this class are self-affine, i.e. have fractal structure. We derive an expression for the definite integral over the area of definition for the functions in this class.

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